The Department of Mathematics and Statistics proposes the following
course for inclusion in University Studies, Unity and Diversity, Critical Analysis at
Winona State University. This was approved by the full department on Thursday, January 4,
2001.

General Discussion of University Studies  Critical Analysis in
relation to MATH 210:

University Studies: Critical Analysis

Critical Analysis courses in the University Studies program are devoted
to teaching critical thinking or analytic problem-solving skills. These skills include the
ability to identify sound arguments and distinguish them from fallacious ones. The
objective of these courses is to develop students abilities to effectively use the process
of critical analysis. Disciplinary examples should be selected to support the development
of critical analysis skills. These courses must include requirements and learning
activities that promote students abilities to

a. evaluate the validity and reliability of information;b.

Information often comes in the form of a claim or theory. It is
unfortunately all too common that when such information appears in print, the public
reaction is to take the truth of such on the faith that those things which are typeset in
the newspaper or in textbooks must be true (especially if the information involved some
sort of mathematics and/or statistics). In MATH 210, students develop the ability to
examine arguments first on the method of proof used  either deductive or inductive
(i.e., based upon experiment)  and then on the symbolic form of that method.
Students study the reliability of inductive reasoning versus deductive reasoning. Further,
through symbolic logic, they learn to systemize their evaluative methods of examining the
validity of deductive arguments. As such, students learn a very powerful mathematical
technique towards evaluating the validity and reliability of claims or theories.

One example of the form a theory might take is "If something
happens, then something will also happen". Although a seemingly clear and simple
promise of when one thing will follow from another, there are several ways such a theory
may be stated equivalently yet appear to say something quite different. Yet another
possibility is that an "If-then" sort of theory may be twisted around a bit,
appear to be equivalent to the original theory, and, in fact, be completely different.
Terminologies associated with such an example are conditional, contrapositive, converse,
and inverse. Students develop the ability to see which of these are equivalent, which are
not, and why. They also learn to recognize the many different forms, involving such terms
as "necessary" and "sufficient", that these may take in everyday print
(with or without mathematics). Students develop the ability to understand the ways in
which these statements may be validly deduced, and they learn the ways in which a faulty
line of reasoning may be turned around to look like it is valid.

d. recognize possible inadequacies or biases in the evidence
given to support arguments or conclusions; ande.

The way in which evidence is assembled and used to argue the truth of a
point may result in either a solid, deductive argument or one of several classic errors in
reasoning. For example, a popular movie some years back included a court case where a key
witness, a colonel as it turns out, claims to have had nothing to do with a certain
soldier's death. Upon cross-examination, he offers three key pieces of testimony. First,
all of his soldiers follow orders. No exceptions. Second, he ordered his soldiers not to
harm another soldier on his base in any way. Third, before the soldier who died met with
his untimely death, this colonel issued him a transfer to another base. The reason given
was that this soldier was substandard, and other soldier may harm the man for not being up
to standards. Given some thought, it is possible to see that the colonel has lied.
Students in MATH 210 develop the ability to apply symbolic logic to clearly recognize the
contradiction in the evidence provided. They then go on to other, more elaborate examples,
to see that such contradictions are quite common occurrences and that, turning the example
around a bit, "proof by contradiction" becomes a very powerful method of
deductive reasoning.

f. advance or support claims.

One of the key goals of MATH 210 is that students learn to develop the
truth of mathematical claims or theories through valid methods of deductive reasoning.
Further, they learn to apply the meaning of quantified statements toward proper reasoning
and also toward discovery of those claims which turn out to be false. Open-ended problems
challenge students to either develop the truth of a claim or show that it is false, and,
ultimately, students develop their own "theorems" as extensions of this work.

Discrete Mathematics and Foundations (MATH 210) 4 s.h.

Course Syllabus/Outline

Course Title: Discrete Mathematics and Foundations MATH 210

Number of Credits: 4 S.H. Frequency of Offering: Every Semester

Prerequisite(s): MATH 110 or MATH 120 or MATH 150

Grading: Grade only for all majors, minors, options, concentrations and
licensures within the Department of Mathematics and Statistics. The P/NC option is
available to others.

Course Description: An introduction to symbolic logic, quantifiers,
arguments and their use in proving theorems. Examples are taken from elementary number
theory, sequences, sets, and everyday situations. This is a University Studies course
satisfying requirements in Critical Analysis.

Statement of major focus and objectives of the course: The major focus
of this course is to provide students with the ability to write proofs, the ability to
read proofs and decide what method is being used to prove the statement, the ability to
apply symbolic logic and methods of reasoning to everyday situations.

Note that a focus of the course will be to prepare students to develop
the competencies outlined in the following Minnesota Standards of Effective Teaching
Practice for Beginning Teachers: Standard 1 -- Subject Matter Objectives

To develop within the future teacher ...

�

the ability to use a problem-solving
approach to investigate and understand mathematical content

�

the ability to formulate and solve
problems from both mathematical and everyday situations

�

the ability to communicate mathematical
ideas in writing, using everyday and mathematical language, including symbols

�

the ability to communicate mathematical
ideas orally, using both everyday and mathematical language

�

the ability to make and evaluate
mathematical conjectures and arguments and validate their own mathematical thinking

�

the ability to connect mathematics to
other disciplines and real-world situations

�

an understanding of and the ability to
apply concepts of number, number theory and number systems

�

the ability to use algebra to describe
patterns, relations and functions and to model and solve problems

University Studies: Critical Analysis

Critical Analysis courses in the University Studies program are devoted
to teaching critical thinking or analytic problem-solving skills. These skills include the
ability to identify sound arguments and distinguish them from fallacious ones. The
objective of these courses is to develop students abilities to effectively use the process
of critical analysis. Disciplinary examples should be selected to support the development
of critical analysis skills. These courses must include requirements and learning
activities that promote students abilities to